Information Geometry and Entropy-Based Inference

Speaker: 

Jun Zhang

Institution: 

University of Michigan

Time: 

Tuesday, March 5, 2019 - 4:00pm

Location: 

RH 306

Information Geometry is the differential geometric study of the manifold of probability models, and promises to be a unifying geometric framework for investigating statistical inference, information theory, machine learning, etc. Instead of using metric for measuring distances on such manifolds, these applications often use “divergence functions” for measuring proximity of two points (that do not impose symmetry and triangular inequality), for instance Kullback-Leibler divergence, Bregman divergence, f-divergence, etc. Divergence functions are tied to generalized entropy (for instance, Tsallis entropy, Renyi entropy, phi-entropy) and cross-entropy functions widely used in machine learning and information sciences. It turns out that divergence functions enjoy pleasant geometric properties – they induce what is called “statistical structure” on a manifold M: a Riemannian metric g together with a pair of torsion-free affine connections D, D*, such that D and D* are both Codazzi coupled to g while being conjugate to each other. Divergence functions also induce a natural symplectic structure on the product manifold MxM for which M with statistical structure is a Lagrange submanifold.  We recently characterize holomorphicity of D, D* in the (para-)Hermitian setting, and show that statistical structures (with torsion-free D, D*) can be enhanced to Kahler or para-Kahler manifolds. The surprisingly rich geometric structures and properties of a statistical manifold open up the intriguing possibility of geometrizing statistical inference, information, and machine learning in string-theoretic languages. 

Nonnegative Ricci curvature, stability at infinity, and structure of fundamental groups

Speaker: 

Jiayin Pan

Institution: 

UCSB

Time: 

Tuesday, February 12, 2019 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

We study the fundamental group of an open n-manifold of nonnegative Ricci curvature with some additional condition on the Riemannian universal cover. We show that if the universal cover satisfies certain geometric stability condition at infinity, the \pi_1(M) is finitely generated and contains an abelian subgroup of finite index. This can be applied to the case that the universal cover has a unique tangent cone at infinity as a metric cone or the case that the universal cover has Euclidean volume growth of constant 1-\epsilon(n).

 

Mass, Kaehler Manifolds, and Symplectic Geometry

Speaker: 

Claude LeBrun

Institution: 

Stony Brook University

Time: 

Tuesday, April 2, 2019 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

In the author's previous joint work with Hans-Joachim Hein, a mass formula for asymptotically locally Euclidean (ALE) Kaehler manifolds was proved, assuming only relatively weak fall-off conditions on the metric. However, the case of real dimension four presented technical difficulties that led us to require fall-off conditions in this special dimension that are stronger than the Chrusciel fall-off conditions that sufficed in higher dimensions. In this talk, I will explain how a new proof of the 4-dimensional case, using ideas from symplectic geometry, shows that Chrusciel fall-off suffices to imply all our main results in any dimension. In particular, I will explain why our Penrose-type inequality for the mass of an asymptotically Euclidean Kaehler manifold always still holds, given only this very weak metric fall-off hypothesis.

A promenade through the isoparametric story

Speaker: 

Quo-Shin Chi

Institution: 

Washington University in St. Louis

Time: 

Tuesday, April 23, 2019 - 4:00pm

Location: 

RH 306

Isoparametric hypersurfaces in the sphere are those whose
principal curvatures are everywhere constant with fixed multiplicities. In
some sense, such hypersurfaces represent the simplest type of manifolds we
can get a handle on. They have rather complicated topology and most of them
are inhomogeneous, and thus they serve as a good testing ground for
constructing examples and counterexamples. The classification of such
hypersurfaces was initiated by E. Cartan around 1938, and the completion of
the last case with four principal curvatures will appear soon in
publication. Since the classification is a long story covering a wide
spectrum of mathematics, I will highlight in this talk the decisive moments
and the key ideas engaged in the intriguing pursuit.

Laplace eigenvalues and minimal surfaces in spheres

Speaker: 

Mikhail Karpukhin

Institution: 

UC Irvine

Time: 

Tuesday, February 5, 2019 - 4:00pm

Location: 

RH 306

The spectrum of the Laplace-Beltrami operator is
one of the fundamental invariants of a Riemannian manifold.
It has many applications, perhaps the most significant is in relation to
minimal surfaces. In the present talk we will show how minimal surfaces
arise in the study of isoperimetric inequalities for Laplace eigenvalues,
the relation that was initially discovered by P. Li and S. T. Yau. We will
present recent results in this direction and discuss connections to other
fields, including algebraic geometry and cobordism theory. The talk is based
on joint works with V. Medvedev, N. Nadirashvili, A. Penskoi and I.
Polterovich.

Colloquium (special time)

Speaker: 

Duong Phong

Institution: 

Columbia University

Time: 

Tuesday, February 26, 2019 - 4:00pm to 5:00pm

Location: 

RH 306

Since the mid 1990’s, the leading candidate for a unified theory of all fundamental physical interactions has been M Theory.

A full formulation of M Theory is still not available, and it is only understood through its limits in certain regimes, which are either one of five 10-dimensional string theories, or 11-dimensional supergravity. The equations for these theories are mathematically interesting in themselves, as they reflect, either directly or indirectly, the presence of supersymmetry. We discuss recent progresses and open problems about two of these theories, namely supersymmetric compactifications of the heterotic string and of 11-dimensional supergravity. This is based on joint work of the speaker with Sebastien Picard and Xiangwen Zhang, and with Teng Fei and Bin Guo.

Volume estimates for tubes around submanifolds using integral curvature bounds

Speaker: 

Yousef Chahine

Institution: 

UC Santa Barbara

Time: 

Tuesday, December 4, 2018 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

We generalize an inequality of E. Heintze and H. Karcher for the volume of tubes around minimal submanifolds to an inequality based on integral bounds for k-Ricci curvature. Even in the case of a pointwise bound this generalizes the classical inequality by replacing a sectional curvature bound with a k-Ricci bound. This work is motivated by the estimates of Petersen-Shteingold-Wei for the volume of tubes around a geodesic and generalizes their estimate. Using similar ideas we also prove a Hessian comparison theorem for k-Ricci curvature which generalizes the usual Hessian and Laplacian comparison for distance functions from a point and give several applications.

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